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3 of 3 incorporated OP's clarification into my answer

I think you are a:


I live in a palace where it's dark inside
No light in there, unless open wide

Teeth live in mouths, which fit this description

Even though I am not a king, sometimes I wear a crown
I am not a queen either, neither I am a clown

Teeth are neither kings, queens, nor clowns, but they do sometimes have crowns

I'm known for tearing stuff by moving up and down

This describes chewing food

Don't take care of me and I might hurt you

If you don't brush and floss your teeth, they'll start to hurt!

Take care of me and get better pictures of you

People like to have a dazzling white smile for photos!


Without teeth, it would be harder to eat, which would definitely make life hard!